Question
If both the roots of the quadratic equation $${x^2} - 2kx + {k^2} + k - 5 = 0$$ are less than 5, then $$k$$ lies in the interval
A.
$$\left( {5,6} \right]$$
B.
$$\left( {6,\infty } \right)$$
C.
$$\left( { - \infty ,4} \right)$$
D.
$$\left[ {4,5} \right]$$
Answer :
$$\left( { - \infty ,4} \right)$$
Solution :
both roots are less than 5 then (i) Discriminant $$ \geqslant 0$$
$$\eqalign{ & \left( {{\text{ii}}} \right)\,\,p\left( 5 \right) > 0 \cr & \left( {{\text{iii}}} \right)\,\,\frac{{{\text{Sum of roots}}}}{2} < 5 \cr} $$
$$\eqalign{ & {\text{Hence }}\left( {\text{i}} \right)4{k^2} - 4\left( {{k^2} + k - 5} \right) \geqslant 0 \cr & 4{k^2} - 4{k^2} - 4k + 20 \geqslant 0 \cr & 4k \leqslant 20 \cr & \Rightarrow \,\,k \leqslant 5 \cr & \left( {{\text{ii}}} \right)\,\,f\left( 5 \right) > 0;25 - 10k + {k^2} + k - 5 > 0 \cr & {\text{or }}\,{k^2} - 9k + 20 > 0 \cr & {\text{or }}k\left( {k - 4} \right) - 5\left( {k - 4} \right) > 0 \cr & {\text{or }}\left( {k - 5} \right)\left( {k - 4} \right) > 0 \cr & \Rightarrow \,\,k \in \left( { - \infty ,4} \right) \cup \left( { - \infty ,5} \right) \cr & \left( {{\text{iii}}} \right)\,\,\frac{{{\text{Sum of roots}}}}{2} = - \frac{b}{{2a}} = \frac{{2k}}{2} < 5 \cr} $$
The interection of (i), (ii) & (iii) gives
$$k \in \left( { - \infty ,4} \right)$$
both roots are less than 5 then (i) Discriminant $$ \geqslant 0$$
$$\eqalign{ & \left( {{\text{ii}}} \right)\,\,p\left( 5 \right) > 0 \cr & \left( {{\text{iii}}} \right)\,\,\frac{{{\text{Sum of roots}}}}{2} < 5 \cr} $$
$$\eqalign{ & {\text{Hence }}\left( {\text{i}} \right)4{k^2} - 4\left( {{k^2} + k - 5} \right) \geqslant 0 \cr & 4{k^2} - 4{k^2} - 4k + 20 \geqslant 0 \cr & 4k \leqslant 20 \cr & \Rightarrow \,\,k \leqslant 5 \cr & \left( {{\text{ii}}} \right)\,\,f\left( 5 \right) > 0;25 - 10k + {k^2} + k - 5 > 0 \cr & {\text{or }}\,{k^2} - 9k + 20 > 0 \cr & {\text{or }}k\left( {k - 4} \right) - 5\left( {k - 4} \right) > 0 \cr & {\text{or }}\left( {k - 5} \right)\left( {k - 4} \right) > 0 \cr & \Rightarrow \,\,k \in \left( { - \infty ,4} \right) \cup \left( { - \infty ,5} \right) \cr & \left( {{\text{iii}}} \right)\,\,\frac{{{\text{Sum of roots}}}}{2} = - \frac{b}{{2a}} = \frac{{2k}}{2} < 5 \cr} $$
The interection of (i), (ii) & (iii) gives
$$k \in \left( { - \infty ,4} \right)$$